The expression provided in the image is:

$\frac{4x^2 - 11x + 6}{16x^2 - 9}$

Step 1: Factor the numerator and the denominator

Numerator: $4x^2 - 11x + 6$

We need to factor this quadratic expression. We look for two numbers that multiply to $4 \times 6 = 24$ and add to $-11$. The numbers are $-3$ and $-8$.

So we rewrite $-11x$ as $-3x - 8x$:

$4x^2 - 11x + 6 = 4x^2 - 3x - 8x + 6$

Now factor by grouping:

$= (4x^2 - 8x) - (3x - 6)$ $= 4x(x - 2) - 3(x - 2)$ $= (4x - 3)(x - 2)$

Denominator: $16x^2 - 9$

This is a difference of squares:

$16x^2 - 9 = (4x - 3)(4x + 3)$

Step 2: Simplify the expression

Now we have:

$\frac{(4x - 3)(x - 2)}{(4x - 3)(4x + 3)}$

Cancel the common factor $(4x - 3)$ from the numerator and the denominator:

$\frac{x - 2}{4x + 3}$

Final Answer:

The simplified form of the expression is:

$\frac{x - 2}{4x + 3}$

Would you like further details or explanations? Here are some related questions:

1. How do you factor a quadratic expression like $4x^2 - 11x + 6$?
2. What is the difference of squares and how is it factored?
3. Why can you cancel out terms like $4x - 3$ in a rational expression?
4. How would the expression change if it were $16x^2 + 9$ instead of $16x^2 - 9$?
5. What if the quadratic in the denominator couldn't be factored? How would you approach it?

Tip: Always check for common factors when simplifying algebraic expressions, but make sure that any term you're canceling is a factor, not just part of a term!